Wave Front Tracking using Template Matching and Segmented Regression

An experimental study was designed to measure shock waves (supersonic gas flow) in a shock tube. A high-speed camera captured images of the shock waves, at a framerate up to 500,000 frames per second. with respect to the huge number of images to be analyzed, an image-processing algorithm was developed for automatic tracking of the shock waves. However, each shock wave might be divided into to two parts; a normal shock (the shock wave is perpendicular to the flow direction), and an oblique shock (the shock is at an oblique angle relative to the flow direction). The proposed framework calculates the characteristics of the wave front, i.e. the angle and velocity of normal and oblique shocks. A technique based on Template Matching and an extended version of Segmented Regression is developed to track the wave front in the high-speed videos. To our understanding, the proposed framework is novel, and our findings are in accordance with results derived from pressure sensors within the test tube

Level Set Method for Positron Emission Tomography

In positron emission tomography (PET), a radioactive compound is injected into the body to promote a tissue-dependent emission rate. Expectation maximization (EM) reconstruction algorithms are iterative techniques which estimate the concentration coefficients that provide the best fitted solution, for example, a maximum likelihood estimate. In this paper, we combine the EM algorithm with a level set approach. The level set method is used to capture the coarse scale information and the discontinuities of the concentration coefficients. An intrinsic advantage of the level set formulation is that anatomical information can be efficiently incorporated and used in an easy and natural way. We utilize a multiple level set formulation to represent the geometry of the objects in the scene. The proposed algorithm can be applied to any PET configuration, without major modifications

Iterative Image Restoration Combining Total Variation Minimization and a Second-Order Functional ∗

Abstract. A noise removal technique using partial differential equations (PDEs) is proposed here. It combines the Total Variational (TV) filter with a fourth-order PDE filter. The combined technique is able to preserve edges and at the same time avoid the staircase effect in smooth regions. A weighting function is used in an iterative way to combine the solutions of the TV-filter and the fourth-order filter. Numerical experiments confirm that the new method is able to use less restrictive time step than the fourth-order filter. Numerical examples using images with objects consisting of edge, flat and intermediate regions illustrate advantages of the proposed model

A Binary Level Set Model and some Applications to Mumford-Shah Image Segmentation

In this work we propose a variant of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can at convergence only take two values, i.e. it can only be 1 or-1. Some of the properties of the standard level set function are preserved in the proposed method, while others are not. Using this new level set method for interface problems, we need to minimize a smooth convex functional under a quadratic constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth and locally convex. We show numerical results using the method for segmentation of digital images

International Conference Imaging, Vision and Learning Based on Optimization and PDEs

This volume presents the peer-reviewed proceedings of the international conference Imaging, Vision and Learning Based on Optimization and PDEs (IVLOPDE), held in Bergen, Norway, in August/September 2016. The contributions cover state-of-the-art research on mathematical techniques for image processing, computer vision and machine learning based on optimization and partial differential equations (PDEs). It has become an established paradigm to formulate problems within image processing and computer vision as PDEs, variational problems or finite dimensional optimization problems. This compact yet expressive framework makes it possible to incorporate a range of desired properties of the solutions and to design algorithms based on well-founded mathematical theory. A growing body of research has also approached more general problems within data analysis and machine learning from the same perspective, and demonstrated the advantages over earlier, more established algorithms. This volume will appeal to all mathematicians and computer scientists interested in novel techniques and analytical results for optimization, variational models and PDEs, together with experimental results on applications ranging from early image formation to high-level image and data analysis

A variant of the level set method and applications to image segmentation

In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of level set functions are utilized to identify up to 2n phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If phases should be identified, the level set function must approach 2n predetermined constants. We just need one level set function to represent 2n unique phases, and this gains in storage capacity. Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the Delta and Heaviside functions. Numerical examples are given, and we also compare our method with related approaches.Published versio

Noise removal using smoothed normals and surface fitting

In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps.We first use a total-variation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoothed normal vectors. For each of these two stages, the problem is reduced to a nonlinear partial differential equation. Finite difference schemes are used to solve these equations. A broad range of numerical examples are given in the paper.Published versio

Computing Ischemic Regions in the Heart: On the Use of Internal Electrodes

In order to better locate ischemic regions in the heart using electrical measurements and inverse solutions, we explore the possibility for supplementing BSPM data sets with additional internal electrodes in the esophagus. We investigated whether such internal electrodes closer to the heart's surface could significantly improve the ability to pinpoint ischemic regions. A framework based on exercise ECG testing and a mathematical model for identifying ischemic regions from ECG measurements was implemented to test the effect of potential internal electrodes. This method identifies areas with abnormal perfusion by minimizing the difference between recorded and simulated ECGs. To investigate the effect of the extra electrodes in the esophagus, we computed the location of the ischemic zones with and without the internal electrodes for both synthetic data and using clinically obtained BSPMs. Computations based on pure synthetic data illuminate that, if an ischemic region is close to an electrode in the esophagus, then the use of internal electrodes might improve the result significantly. However, the simulations also indicate that ischemic areas further away from the internal electrodes are not better recovered with the use of such additional ECGs. This study indicates that the use internal electrodes, along with standard BSPMs, might improve the accuracy of the inverse ECG technology